# Theory Closures

```(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)
theory Closures
imports Myhill "HOL-Library.Infinite_Set"
begin

section ‹Closure properties of regular languages›

abbreviation
regular :: "'a lang ⇒ bool"
where
"regular A ≡ ∃r. A = lang r"

subsection ‹Closure under ‹∪›, ‹⋅› and ‹⋆››

lemma closure_union [intro]:
assumes "regular A" "regular B"
shows "regular (A ∪ B)"
proof -
from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto
then have "A ∪ B = lang (Plus r1 r2)" by simp
then show "regular (A ∪ B)" by blast
qed

lemma closure_seq [intro]:
assumes "regular A" "regular B"
shows "regular (A ⋅ B)"
proof -
from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto
then have "A ⋅ B = lang (Times r1 r2)" by simp
then show "regular (A ⋅ B)" by blast
qed

lemma closure_star [intro]:
assumes "regular A"
shows "regular (A⋆)"
proof -
from assms obtain r::"'a rexp" where "lang r = A" by auto
then have "A⋆ = lang (Star r)" by simp
then show "regular (A⋆)" by blast
qed

subsection ‹Closure under complementation›

text ‹Closure under complementation is proved via the
Myhill-Nerode theorem›

lemma closure_complement [intro]:
fixes A::"('a::finite) lang"
assumes "regular A"
shows "regular (- A)"
proof -
from assms have "finite (UNIV // ≈A)" by (simp add: Myhill_Nerode)
then have "finite (UNIV // ≈(-A))" by (simp add: str_eq_def)
then show "regular (- A)" by (simp add: Myhill_Nerode)
qed

subsection ‹Closure under ‹-› and ‹∩››

lemma closure_difference [intro]:
fixes A::"('a::finite) lang"
assumes "regular A" "regular B"
shows "regular (A - B)"
proof -
have "A - B = - (- A ∪ B)" by blast
moreover
have "regular (- (- A ∪ B))"
using assms by blast
ultimately show "regular (A - B)" by simp
qed

lemma closure_intersection [intro]:
fixes A::"('a::finite) lang"
assumes "regular A" "regular B"
shows "regular (A ∩ B)"
proof -
have "A ∩ B = - (- A ∪ - B)" by blast
moreover
have "regular (- (- A ∪ - B))"
using assms by blast
ultimately show "regular (A ∩ B)" by simp
qed

subsection ‹Closure under string reversal›

fun
Rev :: "'a rexp ⇒ 'a rexp"
where
"Rev Zero = Zero"
| "Rev One = One"
| "Rev (Atom c) = Atom c"
| "Rev (Plus r1 r2) = Plus (Rev r1) (Rev r2)"
| "Rev (Times r1 r2) = Times (Rev r2) (Rev r1)"
| "Rev (Star r) = Star (Rev r)"

lemma rev_seq[simp]:
shows "rev ` (B ⋅ A) = (rev ` A) ⋅ (rev ` B)"
unfolding conc_def image_def
by (auto) (metis rev_append)+

lemma rev_star1:
assumes a: "s ∈ (rev ` A)⋆"
shows "s ∈ rev ` (A⋆)"
using a
proof(induct rule: star_induct)
case (append s1 s2)
have inj: "inj (rev::'a list ⇒ 'a list)" unfolding inj_on_def by auto
have "s1 ∈ rev ` A" "s2 ∈ rev ` (A⋆)" by fact+
then obtain x1 x2 where "x1 ∈ A" "x2 ∈ A⋆" and eqs: "s1 = rev x1" "s2 = rev x2" by auto
then have "x1 ∈ A⋆" "x2 ∈ A⋆" by (auto)
then have "x2 @ x1 ∈ A⋆" by (auto)
then have "rev (x2 @ x1) ∈ rev ` A⋆" using inj by (simp only: inj_image_mem_iff)
then show "s1 @ s2 ∈  rev ` A⋆" using eqs by simp
qed (auto)

lemma rev_star2:
assumes a: "s ∈ A⋆"
shows "rev s ∈ (rev ` A)⋆"
using a
proof(induct rule: star_induct)
case (append s1 s2)
have inj: "inj (rev::'a list ⇒ 'a list)" unfolding inj_on_def by auto
have "s1 ∈ A"by fact
then have "rev s1 ∈ rev ` A" using inj by (simp only: inj_image_mem_iff)
then have "rev s1 ∈ (rev ` A)⋆" by (auto)
moreover
have "rev s2 ∈ (rev ` A)⋆" by fact
ultimately show "rev (s1 @ s2) ∈  (rev ` A)⋆" by (auto)
qed (auto)

lemma rev_star [simp]:
shows " rev ` (A⋆) = (rev ` A)⋆"
using rev_star1 rev_star2 by auto

lemma rev_lang:
shows "rev ` (lang r) = lang (Rev r)"
by (induct r) (simp_all add: image_Un)

lemma closure_reversal [intro]:
assumes "regular A"
shows "regular (rev ` A)"
proof -
from assms obtain r::"'a rexp" where "A = lang r" by auto
then have "lang (Rev r) = rev ` A" by (simp add: rev_lang)
then show "regular (rev` A)" by blast
qed

subsection ‹Closure under left-quotients›

abbreviation
"Deriv_lang A B ≡ ⋃x ∈ A. Derivs x B"

lemma closure_left_quotient:
assumes "regular A"
shows "regular (Deriv_lang B A)"
proof -
from assms obtain r::"'a rexp" where eq: "lang r = A" by auto
have fin: "finite (pderivs_lang B r)" by (rule finite_pderivs_lang)

have "Deriv_lang B (lang r) = (⋃ (lang ` pderivs_lang B r))"
also have "… = lang (⨄(pderivs_lang B r))" using fin by simp
finally have "Deriv_lang B A = lang (⨄(pderivs_lang B r))" using eq
by simp
then show "regular (Deriv_lang B A)" by auto
qed

subsection ‹Finite and co-finite sets are regular›

lemma singleton_regular:
shows "regular {s}"
proof (induct s)
case Nil
have "{[]} = lang (One)" by simp
then show "regular {[]}" by blast
next
case (Cons c s)
have "regular {s}" by fact
then obtain r where "{s} = lang r" by blast
then have "{c # s} = lang (Times (Atom c) r)"
then show "regular {c # s}" by blast
qed

lemma finite_regular:
assumes "finite A"
shows "regular A"
using assms
proof (induct)
case empty
have "{} = lang (Zero)" by simp
then show "regular {}" by blast
next
case (insert s A)
have "regular {s}" by (simp add: singleton_regular)
moreover
have "regular A" by fact
ultimately have "regular ({s} ∪ A)" by (rule closure_union)
then show "regular (insert s A)" by simp
qed

lemma cofinite_regular:
fixes A::"'a::finite lang"
assumes "finite (- A)"
shows "regular A"
proof -
from assms have "regular (- A)" by (simp add: finite_regular)
then have "regular (-(- A))" by (rule closure_complement)
then show "regular A" by simp
qed

subsection ‹Continuation lemma for showing non-regularity of languages›

lemma continuation_lemma:
fixes A B::"'a::finite lang"
assumes reg: "regular A"
and     inf: "infinite B"
shows "∃x ∈ B. ∃y ∈ B. x ≠ y ∧ x ≈A y"
proof -
define eqfun where "eqfun = (λA x::('a::finite list). (≈A) `` {x})"
have "finite (UNIV // ≈A)" using reg by (simp add: Myhill_Nerode)
moreover
have "(eqfun A) ` B ⊆ UNIV // (≈A)"
unfolding eqfun_def quotient_def by auto
ultimately have "finite ((eqfun A) ` B)" by (rule rev_finite_subset)
with inf have "∃a ∈ B. infinite {b ∈ B. eqfun A b = eqfun A a}"
by (rule pigeonhole_infinite)
then obtain a where in_a: "a ∈ B" and "infinite {b ∈ B. eqfun A b = eqfun A a}"
by blast
moreover
have "{b ∈ B. eqfun A b = eqfun A a} = {b ∈ B. b ≈A a}"
unfolding eqfun_def Image_def str_eq_def by auto
ultimately have "infinite {b ∈ B. b ≈A a}" by simp
then have "infinite ({b ∈ B. b ≈A a} - {a})" by simp
moreover
have "{b ∈ B. b ≈A a} - {a} = {b ∈ B. b ≈A a ∧ b ≠ a}" by auto
ultimately have "infinite {b ∈ B. b ≈A a ∧ b ≠ a}" by simp
then have "{b ∈ B. b ≈A a ∧ b ≠ a} ≠ {}"
by (metis finite.emptyI)
then obtain b where "b ∈ B" "b ≠ a" "b ≈A a" by blast
with in_a show "∃x ∈ B. ∃y ∈ B. x ≠ y ∧ x ≈A y"
by blast
qed

subsection ‹The language ‹a⇧n b⇧n› is not regular›

abbreviation
replicate_rev ("_ ^^^ _" [100, 100] 100)
where
"a ^^^ n ≡ replicate n a"

lemma an_bn_not_regular:
shows "¬ regular (⋃n. {CHR ''a'' ^^^ n @ CHR ''b'' ^^^ n})"
proof
define A where "A = (⋃n. {CHR ''a'' ^^^ n @ CHR ''b'' ^^^ n})"
assume as: "regular A"
define B where "B = (⋃n. {CHR ''a'' ^^^ n})"

have sameness: "⋀i j. CHR ''a'' ^^^ i @ CHR ''b'' ^^^ j ∈ A ⟷ i = j"
unfolding A_def
apply auto
apply(drule_tac f="λs. length (filter ((=) (CHR ''a'')) s) = length (filter ((=) (CHR ''b'')) s)"
in arg_cong)
apply(simp)
done

have b: "infinite B"
unfolding infinite_iff_countable_subset
unfolding inj_on_def B_def
by (rule_tac x="λn. CHR ''a'' ^^^ n" in exI) (auto)
moreover
have "∀x ∈ B. ∀y ∈ B. x ≠ y ⟶ ¬ (x ≈A y)"
apply(auto)
unfolding B_def
apply(auto)